Analytic Solution for 1D Transient Heat Conduction
The problem geometry and boundary conditions are shown below. An initially isothermal (Tinitial) semi-infinite medium is suddenly subject to a surface temperature Th.
The temperature field can be non-dimensionalized as:
The governing differential equation (with spatially one-dimensional heat flow) is
The solution for all locations x and times t is:
where α is the material’s thermal diffusivity.
Graphical CDF Tool
The following is an embedded, active Mathematica CDF tool. The units for α are cm2/sec, with corresponding units of cm and sec for x and t, respectively.